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Chapter 30 The Production and Properties
of Magnetic fields
In chapter 29 we studied the magnetic force FB exerted by a
magnetic field B on a charge q moving with velocity v
In this chapter we examine how magnetic fields are generated
(the answer is: by currents). The connection between B and I
is given by two equivalent laws:
Ampere’s law and the law of Biot-Savart
Ampere’s law is the analog of Gauss’s law for E
The law of Biot-Savart is the analog of Coulomb’s law
(30-1)
Oersted demonstrated that there is a connection between
electric and magnetic phenomena by showing that a wire
that carries a current deflects the magnetic compass. The
current I in the wire produces a magnetic field B that
deflects the compass needle
I=0
I≠0
I
(30-2)
Hans Christian Oersted
1777-1851
Magnetic field B created by a long wire that carries a current I
The magnetic field lines form circles around the wire of radius
r. The magnitude of B depends on I and on r. The direction of
B is given by the following rule:
• Point the thumb of the right hand in
the direction of I
• Curl the fingers of the the right
hand around the wire
• The fingers point in the direction of
the magnetic field B
(30-3)
When two parallel wires of
length L carry currents I1
and I2 in the same direction
they exert on each other
attractive forces given by:
CI1 I2 L
F1 = F2 =
d
(30-4)
When the currents flow in
opposite directions the forces
are repulsive and their
magnitudes are given by the
same expression:
CI1 I2 L
F1 = F2 =
d
(30-5)
ur
ur ur
Force on wire 2 F 2 = I 2 L2 × B1
(
)
→ F2 = I 2 L2 B1 sin(90°) →
F2 = I 2 L2 B1 (eqs.1)
CI1 I 2 L2
Force on wire 2 is given by: F2 =
d
(eqs.2)
CI1 I 2 L2
We compare eqs.1 with eqs.2 → I 2L2 B1 =
→
d
CI1
B1 =
d
d
L2
(30-6)
Choice of the constant C in the SI system of units
CI1
B1 =
The value of C is arbitrarily chosen
d
C = 2 × 10−7 Tm / A
The constant C is written in the form:
µo
C=
→ µ o = 2π C = 1.26 ×10 −6 Tm / A
2π
The expressions for B1 and F2 become:
µo I1
B1 =
2π d
and
µo I1 I2 L2
F2 =
2π d
d
L2
(30-7)
(30-8)
The magnetic force between two current
carrying parallel wires of length L
placed at a distance d can be used to
define the current unit (the Ampere )
Definition of the Ampere (SI current unit)
µ o I1 I2 L
F1 = F2 =
µo = 1.26 ×10−6 Tm / A
2π d
If we set L = 1 m and d = 1 m we get:
F1 = F2 = 2 ×10−7 N
This is the practical definition for the current unit
(the Ampere) in the SI system of units
Path γ
y
∆SN
∆S2
∆S1
C
B2
BN
A
B1
O
x
(30-9)
Line Integral We are given
the magnetic field B, the path
γ, and the start and end points
A and C. Determine:
C ur
ur
∫ B ⋅ d S along the path γ
A
• Divide the path into N elements ∆S1, ∆S2, …, ∆SN
• Determine B 1 . ∆S1 , B 2 . ∆S2 , … , BN . ∆SN
• Sum all N terms and take the limit as N → ∞
C ur
ur
ur
• B ⋅ d Sur =lim  uBr1 ⋅ ∆ Sur1 + uBr 2 ⋅ ∆ uSr 2 + ... + B
N ⋅ ∆S N 
∫

 as N → ∞
A
µo I
B =
2π r
ur ur
Calculate ∫ B ⋅ d S over a circle of
I
B
dS
r
B
dS B
→
dS
radius r for a very long wire that
carries a current I
ur ur
∫ B ⋅ d S = ∫ BdS cos 0 = B ∫ dS = 2π rB
µo I
B =
2π r
→
ur ur
µo I
∫ B ⋅ d S = 2π r 2π r =µ o I
ur ur
∫ B ⋅ d S = µo I
This results is true for any closed path
(30-10)
γ
n
I3
(30-11)
For any closed path γ
ur ur
∫ B ⋅ d S = µo I enc
This equation is known as
“Ampere’s law” and is
one of Maxwell’s equation.
I1
Note 1:
I4
I2
Ienc = I1 - I2
I1 is parallel to n.
I2 is antiparallel to n
Note 2: I3 and I4 are not included in Ienc because they lie
outside γ
Note 3: In problems we try to choose a convenient path γ
Andre Marie Ampere
1775-1836
γ
ur ur
∫ B ⋅ d S = µo I enc
Ampere’s law
Ampere’s law is the analog of Gauss’s law in magnetism. It
is quite useful in situations that have high symmetry. In these
cases Ampere’s law can be used to solve the problem with
minimum effort.
How do we handle problems without symmetry?
In these cases we use the law of Biot-Savart and the principle
of superposition. Biot-Savart law is the analog of Coulomb’s
law in magnetism.
(30-12)
Example 30-1:
Find B inside and outside a
long wire of radius R that
carries a current I
We will apply Ampere’s law
for the two circular paths
indicated in the figures.
One path lies outside the
wire (r > R)
The other lies inside the wire
(r < R)
(30-13)
B outside the wire (r > R)
Apply Ampere’s law on path a
ur ur
∫ B ⋅ d S = ∫ BdS cos 0 = ∫ BdS
∫ BdS = B ∫ dS = 2π rB
ur ur
→ ∫ B ⋅ d S = 2π rB
From Ampere's law:
ur ur
∫ B ⋅ d S = µo Ienc = µo I
→ 2π rB = µ o I
→
µo I
B=
2π r
(30-14)
B inside the wire . Apply
Ampere’s law for path (b)
ur ur
∫ B ⋅ d S = ∫ BdS cos 0 = ∫ BdS
∫ BdSur = Bur∫ dS = 2π rB
→ ∫ B ⋅ d S = 2π rB
From Ampere's law:
ur ur
∫ B ⋅ d S = µo Ienc
π r2
r2
I enc = I
=I 2
2
πR
R
2
r
µ o Ir
2π rB = µ o I 2 → B =
R
2π R 2
(30-15)
Summary
(30-16)
µo I
B=
2π r
outside,
and
µo Ir
B=
2π R 2
µo I
Note: At r = R both solutions give: B =
2π R
inside
Electric field lines terminate on electric charges. There is no
such thing as “magnetic charge” (also known as a “magnetic
monopole”). For this reason the magnetic field lines are
closed curves
(30-17)
Consider the flux of B through
the closed surface S
ur ur
ΦB = ∫ B ⋅ d A
ΦB is proportional to the
number of magnetic field lines
that exit S. The number of
magnetic field lines that exit S is
equal to the number of magnetic
field lines entering S.
Thus ΦB = 0
This result is general and is
known as “Gauss’s law for
magnetism”
(30-18)
S
Carl Friedrich Gauss
1777-1855
(30-19)
Gauss’s law for magnetism
The magnetic flux ΦB for
any closed surface S is zero
S
ur ur
∫ B⋅d A = 0
S
Note: Gauss’s law for the
electric field has the form:
ur ur q
S
E
⋅
d
A
=
∫S
εo
There is no such thing as
“magnetic charge”
Units of ΦB : T.m2 ≡ Weber
(Wb)
A solenoid is an electrical element that generates a uniform
magnetic field. It is the magnetic equivalent of the parallel
plate capacitor which generates a uniform electric field. A
solenoid consists of a coil of wire wound uniformly into a long
cylinder
(30-20)
The magnetic field
lines inside a solenoid
are parallel to the
solenoid axis. Their
direction is found as
follows:
• Curl the fingers of
the right hand in the
direction of I
• the thumb points
along the direction of
the magnetic field B
(30-21)
B
B
B
(30-22)
Calculate
abcda
∫
ur ur
B ⋅d S =
∫
ur ur
B ⋅d S =
∫
ur ur
B⋅dS =
a →b
c →d
d →a
∫
ur ur
B ⋅ d S for the path abcda. Break the path into four parts
∫
BdS cos(90°) = 0
∫
BdS cos(90°) = 0
a →b
∫
ur ur
B⋅dS = 0
b→ c
c→d
∫
d →a
BdS cos(0) = Bl →
∫
abcda
ur ur
B ⋅ d S = Bl
(B = 0)
B
(30-23)
∫
N turns
ur ur
B ⋅ d S =Bl
From Ampere's law:
abcda
I enc = NI
∫
ur ur
B ⋅ d S =µo I enc
abcda
→
Bl = µo NI
µo NI
→ B=
l
N
≡ n (number of turns per unit length)
l
→ B = µo nI
The Torroidal Coil
(N turns) Apply
Ampere’s law for
the circular path
(radius R’) shown in
the figure
th
pa
B
(30-24)
dS
ur ur
'
∫ B ⋅ d S = ∫ BdS cos 0 = B ∫ dS = 2π R B = µo I enc
I enc = NI
→
2π R ' B = µo NI
µ o NI
→ B=
2π R '
Law of Biot-Savart
The law of Biot-Savart gives us the
magnetic field dB at point P generated
by element dl
r r
ur µo I ( d l × r )
dB =
3
4π r
We handle problems with low
symmetry using the law of Biot-Savart
and the principle of superposition
(30-25)
Jean Baptiste Biot
1774-1862
Example 30-5 Find B at the center
of a ring of radius R that carries a
current I. The element dl creates a
magnetic field dB at C (r = R)
(30-26)
r r
ur µ I (d l × r )
dB = o
4π r 3
C
µo Id lR sin(90) µo Id l
dB =
=
3
4π R
4π R 2
µo I
µo I 2π R µo I
dB =
dl =
=
2 ∫
2
4π R
4π R
2π R
γ
The complete story about
Ampere’s law
S
ur ur
∫ B ⋅ d S =µo I enc
Up to this point we have written Ampere’s law as follows:
γ
A second term was added by Maxwell in 1865 that
completed Ampere’s law which is now written as:
ur ur
dΦE
∫γ B ⋅ d S =µo Ienc + µoε o dt
ur ur
where Φ E = ∫ E ⋅ d A
(30-27)
S
Why the fuss about the new term in Ampere’s law?
Maxwell combined the following four equations (now referred
to as “Maxwell’s equations” ):
Gauss’ law for E
Gauss’ law for B
Ampere’s law (with the new term)
Faraday’s law
He combined these equations and got solutions that are waves
which travel in vacuum with speed:
1
v=
= 3 × 108 m / s = speed of light
ε o µo
(30-28)
James Clerk Maxwell
1831-1879
Friedrich Rudolf Hertz
1857-1894